Learning to differentiate

Artificial neural networks together with associated computational libraries provide a powerful framework for constructing both classification and regression algorithms. In this paper we use neural networks to design linear and non-linear discrete differential operators. We show that neural network based operators can be used to construct stable discretizations of initial boundary-value problems by ensuring that the operators satisfy a discrete analogue of integration-by-parts known as summation-by-parts. Our neural network approach with linear activation functions is compared and contrasted with a more traditional linear algebra approach. An application to overlapping grids is explored. The strategy developed in this work opens the door for constructing stable differential operators on general meshes.